# Reference about equivalent form of the Riemann hypothesis

I saw a statement about the Riemann hypothesis in Wikipedia, stating the following:

$$\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=\frac{1}{\zeta(s)}$$ holds for $$Re(s)>\frac{1}{2}$$ is equivalent to the Riemann hypothesis.

I tried to find a reference about it but I can't find it. May anyone provide a reference to the statement(with a proof) to it?

• There are many references for statements which are equivalent to RH. Did you already start searching yourself? Commented Mar 10, 2023 at 13:46
• I tried to use keywords like "Riemann hypothesis", "Dirichlet series" but did not find anything about the proof of the statement. I guess I am looking for a book which proves it. Commented Mar 10, 2023 at 13:52
• Basically it is equivalent to the fact that the Merten's function $M(x) := \sum_{n\leq x}\mu(n)$ is $O(x^{1/2+\varepsilon})$, which is equivalent to RH by elementary facts about Dirichlet series. Commented Mar 10, 2023 at 14:04
• Thanks. I will look into those. Commented Mar 10, 2023 at 14:09

The paper Riemann hypothesis equivalences, Robin inequality, Lagarias criterion, and Riemann hypothesis lists $$36$$ equivalent statements to RH, with references to complete proofs. We have the following statement on page $$8$$:
Equivalence 8: RH is equivalent to the fact that the following series converges for $$ℜ(s) > \frac{1}{2}$$, see [48]: $$\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.$$ And $$[48]$$ is the book by Titchmarsh, The theory of the Riemann zeta function,second ed., Clarendon press. Oxford University press, New York, 1986.